The water wave problem refers to the motion of a free surface over a body of water of finite of infinite depth.
Its classical formulation usually assumes that the fluid is inviscid and irrotational. It is well known since the seminal work of [1]}
that, in this setting, the water wave equations can be written as a Hamiltonian system with a standard Darboux symplectic structure
whose Hamiltonian is the total energy. The canonical conjugate variables are given by \((\eta ,\xi )\) ,
where \(\eta (x,t)\)  is the surface elevation and \(\xi (x,t)\)  denotes the boundary values of the velocity potential on the free surface.
With introduction of the Dirichlet–Neumann operator \(G(\eta ) \)  that maps the Dirichlet boundary condition for a harmonic function
in the fluid domain to its Neumann boundary condition, this Hamiltonian takes an explicit lower-dimensional form
\(H(\eta ,\xi ) = \frac{1}{2} \int _{\mathbb {R}} \Big [ \xi G(\eta )\xi + g \eta ^2 \Big ] dx \,,\) 
